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Homework Help: Group theory

  1. Jan 20, 2008 #1
    [SOLVED] group theory

    1. The problem statement, all variables and given/known data
    Let [itex]\phi:G \to G'[/itex] be a group homomorphism. Show that if |G| is finite, then [itex]|\phi(G)|[/itex] is finite and is a divisor of |G|.


    2. Relevant equations



    3. The attempt at a solution
    Should the last word be |G'|? Then it would follow from Lagrange's Theorem.
     
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  3. Jan 20, 2008 #2

    Hurkyl

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    Nope; it's right as stated. (And can also use Lagrange's theorem in its proof)
     
  4. Jan 20, 2008 #3

    quasar987

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    Think first isomorphism theorem.
     
  5. Jan 20, 2008 #4
    I haven't gotten to the first isomorphism theorem yet, but I don't even need it:

    We know that [itex]\phi^{-1}(\phi(a)) = aH = Ha[/itex], where H = Ker(phi). So, the cardinality of phi(G) will be the index of H in G, which must divide |G| by Lagrange's Theorem.

    Is that right?
     
  6. Jan 20, 2008 #5
    Bingo.
     
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